#### Abstract

We introduce the and vector spaces of holomorphic functions defined in the unit ball of , generalizing previous work like Ouyang et al. (1998), Stroethoff (1989), and Choa et al. (1992). Likewise, we characterize those spaces in terms of harmonic majorants as a generalization of Arellano et al. (2000).

#### 1. Introduction

##### 1.1. Preliminaries in One Complex Variable

Let be the open unit disk in the complex plane . For , let be the Möbius transformation defined by For , we denote Green's function of with logarithmic singularity at by The Bloch space is defined as the set of analytic functions , such that

For , Aulaskari and Lappan [1] introduced in 1994 the spaces as the family of analytic functions satisfying For , this definition was extended in [2].

Theorem 1 (see [2]). *Let and be an analytic function. Then, if and only if
*

With the aim of generalizing and enclosing several weighted function spaces, Zhao in [3] introduced the spaces in the next way.

Let ,, , and be an analytic function. We will say that belongs to if satisfies the integral condition So far, Theorem 1 is generalized for functions of by the following.

Theorem 2 (see [3]). *Let , , , and be an analytic function. Then, if and only if
*

Zhao has shown that for certain intervals of , , and , makes as a Banach space.

In this paper, we present the spaces of holomorphic functions in the unit ball of , that generalize the spaces introduced by Zhao in [3] for analytic functions in the unit disk. At the same time, this work generalizes, mainly, several results of Ouyang et al for -holomorphic functions appearing in [4]. However, the techniques, the methods, and the structure of our work results are completely different to the quoted reference. At the beginning of Section 2, we introduce the spaces and , that is, when we use in the integral representation, the invariant Green function, or when we use the biholomorphism . We remark that the generalization to several complex variables of requires, for and , different intervals to those used in the one-dimensional case. In Theorems 12 and 13, we draw attention to the continuity of the integral expressions defining these spaces to conclude easily the inclusions of the little classes and in and , respectively.

In Section 2.2, we present what is the most natural Bloch space associated to and several characterizations that we give in Theorem 22. It is important to compare this statement with the results of [4, Proposition 3.6] and [5, Theorem 2.4]. In Section 3, we present in Corollary 30 the equivalence between and , generalizing Proposition 3.4 of [4] and Theorem 1 of [3]. Finally in Section 4, we obtain the characterizations of and in terms of harmonic majorants. Giving in this way, the holomorphic version of the results of Arellano et al. in [6].

##### 1.2. Preliminaries in Several Variables

Consider the vector space with the Hermitian product defined as and the Euclidean norm on defined as

The open ball with center and radius denoted by is defined as and its closure is We will denote , , and .

The function , of class , is harmonic in if for all . Moreover, results harmonic in if and only if

Let . The Möbius function is defined as with and , for , and define (see [7]).

This set of functions is a part of the family of automorphisms on denoted by .

Lemma 3 (see [7, 8]). *The function satisfies the following properties:*(i)*;*(ii)*;*(iii)*the real Jacobian for in is given by
*

Lemma 4. *The real Jacobian of satisfies
*

*Proof. *It follows directly from .

Define, for and ,

##### 1.3. Invariant Gradient and Measure

The collection of all holomorphic functions in is denoted by . Let denote the complex gradient of and denote the radial derivative of . Let , where is the normalized volume measure in . Then, is a Möbius invariant; that is for each -integrable function and .

Let denote the invariant Laplacian of [8], and let denote the invariant gradient of [7]. It is well known [7] that, for , where . In [7], the invariant Green's function is defined as , where As in [7], the -dimensional invariant Bloch space is the set of holomorphic functions , such that their Bloch invariant norm satisfies this is equivalent [4, Lemma 3.2] to, say,

Proposition 5 (see [7]). *Let be an integer; then there are positive constants and , such that, for all ,
*

Proposition 6 (see [9]). *For , one has
*

Theorem 7 (see [7, Theorem 1.12]). *Suppose that is real and . Then, the integral
**
has the following asymptotic properties.*(i)*If , is bounded in .*(ii)*If , then
as .*(iii)*If , then
as .*

Let , . We say that , , and satisfy condition (A) if and satisfy condition (B) if

Proposition 8. *Suppose that , , and satisfy condition (A). Then,
*

*Proof. *We have the following equalities:
The singularity at 0 is integrable if . Now, if we estimate
as if and , , then the result follows from Theorem 7 for any case of .

In a similar way, we have the following.

Proposition 9. *Suppose that , , and satisfy condition (B). Then,
*

#### 2. Function Spaces and

##### 2.1. Main Definitions

In this section, we define the spaces , and their little associated spaces. We study some properties and relations between them.

Suppose that , and satisfy condition (A), and let . Define Suppose that , , and satisfy condition (B), and let . Define It is clear that Associated to these functions, we define the following vector spaces of holomorphic functions. (i)Suppose that , , and satisfy condition (A). belongs to if it satisfies and belongs to the little space if it satisfies (ii)Suppose that , , and satisfy condition (B). belongs to if it satisfies and belongs to the little space if it satisfies (iii)Let . belongs to the Dirichlet space if

As , then for , the sets and are normed vector function spaces, with norms given by This is a direct consequence of the usual theory when we consider the measures ,, respectively. For and nonnegative , , we have ; as a consequence, the sets and result in metric vector spaces, with the metric given by respectively (see [10]). We will prove their completeness in Theorem 26.

*Example 10. *Let denotes the class of holomorphic functions on a neighborhood of . Let . Suppose that , , and satisfy condition (A). By (19) we have, for example, the estimation
So, in the definitions, the parameters , , and were chosen to guarantee that the vector spaces and include at least this wide class of holomorphic functions, as Propositions 5, 8, and 9 show. Compare this example with Proposition 3.7 and Theorem 3.8 from [4].

Let , with power series development given by
such that . Then, belongs to each one of the previous spaces. By (37), we get the same result for and .

The spaces are Möbius invariant in the following sense.

Theorem 11. *Suppose that , , and satisfy condition (A). If and is a Möbius transformation, then .*

*Proof. *Let ,, and . By the change of variables formula and the Möbius invariance of , and , we have
where we have used in the last line the estimations of Lemmas 3 and 4.

Observe that if and , then and these spaces are Möbius invariant in the classical way.

Theorem 12. *Suppose that , , and satisfy condition (A). Suppose that
**
for all . Then, the function is continuous on . *

*Proof. *Let be fixed and , such that . Consider a neighborhood with . Let be a sequence contained in and convergent to . Consider now and define . Then, is bounded in . Thus, by bounded convergence theorem,
converges to
Now, integrating in and using change of variables,

Consider
and define the measure by
Now, if we take , , then . Besides, on , results are bounded by a constant , because . Now,
where denotes the characteristic function of the set . As
then by Lebesgue’s dominated convergence theorem, we have that (55) converges to

Theorem 13. *Suppose that , , and satisfy condition (B). Suppose that
**
for all . Then, the function is continuous on . *

*Proof. *Let and , such that . If , the function defined by
results to be uniformly continuous on . As , then
If is nonconstant, given that , there exists , such that and ; then
Thus, if , then

*Remark 14. *Last two theorems are the versions in of Theorem 3.1 of [6].

Some relations between the little spaces , and the spaces , are given in the following results.

Proposition 15. *Suppose that , , and satisfy condition (A). If , then the function
**
is continuous on .*

*Proof. *By the Theorem 12, it suffices to prove that if , given , there exists such that if , then ; that is, . By hypothesis, . So, given , there exists , such that if , then , or implies . Moreover, as is compact, results uniformly continuous on .

Proposition 16. *Suppose that , , and satisfy condition (A). Then, is contained in .*

*Proof. *In fact if , as the function is uniformly continuous on , then is bounded and so

With the same arguments, we get .

##### 2.2. Characterization as Bloch Spaces

Now, we are going to present several characterizations of Bloch spaces similar to the characterizations obtained by Stroethoff [11, Theorems 4 and 7] and Zhao [3, Theorem 1].

For , the -Bloch spaces, , are the families of holomorphic functions satisfying It is easy to verify that is a norm on the vector space and contains at least the holomorphic functions on domains , with . The respective little -Bloch spaces satisfy It is clear that .

Proposition 17. *
(i) Suppose that , , and satisfy condition (A). Then,
**
and there exists , such that
**
(ii) If . Then,
*

*Proof. *(i) Let ; then by Propositions 5 and 6 and (20)
where we have used that is a subharmonic function; (i) follows from this estimation.

(ii) Let and be fixed. By the absolute continuity of the integral and by Lemma 3, given , there exists , such that if we choose sufficiently close to 1 then

With the same kind of arguments used in (i), we obtain
for all , sufficiently close to 1, and we conclude the proof.

The following result says that Dirichlet spaces are the limit case of and , when .

Theorem 18. *
(i) Suppose that , , and satisfy condition (A). Then,
**
(ii) Suppose that , , and satisfy condition (B). Then,
*

*Proof. *We prove the first inclusion. The proof of the second one is similar. Let and . By Proposition 17, there exists , such that
Let , such that if ; then . Thus,
By Proposition 8, the first integral goes to zero as . For the second one, consider (75) as following:
Moreover, for the third, using the change of variable ,
and this concludes the proof.

Proposition 19. *Suppose that , , and satisfy condition (A), with . Then,
*

*Proof. *Let . Then,
Using change of variables and by Proposition 5 we have
and this integral results convergent because
when . Finally, if .

Similar results to Propositions 17 and 19 can be obtained for .

Proposition 20. *Suppose that , , and satisfy condition (B), with . Then,
**
and there exists , such that
*

*Proof. *Let ; then
and we finish the proof as in the Proposition 17.

Proposition 21. *Suppose that , , and satisfy condition (A), with . Then,
*

*Proof. *Let . By hypothesis,
Thus,
and by Theorem 7, we obtain that if ,
and the result follows.

We can synthesize the last propositions in the following.

Theorem 22. *Suppose that , , and satisfy condition (B). The following statements are equivalent.*(i)*;*(ii)* for each ;*(iii)* for some ;*(iv)* for each ;*(v)* for some .*

Jointly with these properties, there are similar results for the little spaces , , and .

For these cases, the inclusions follow immediately from the previous estimations. For the other inclusions we have the following.

Proposition 23. *Suppose that , , and satisfy condition (A) and . Then,
*

*Proof. *Since , then . Suppose that ; then
Given , there exists , such that
if . We split the integral
as
By hypothesis, the second integral satisfies
Now,